Counting Meromorphic Functions with Critical Points of Large Multiplicities

We study the number of meromorphic functions on a Riemann surface with given critical values and prescribed multiplicities of critical points and values. When the Riemann surface is CP1 and the function is a polynomial, we give an elementary way of finding this number. In the general case, we show that, as the multiplicities of critical points tend to infinity, the asymptotic for the number of meromorphic functions is given by the volume of some space of graphs glued from circles. We express this volume as a matrix integral. 1 From functions to constellations In this section we sketch the more or less standard construction that allows one to reduce the enumeration problem of meromorphic functions to a combinatorial problem of enumerating constellations. In the sequel we will be concerned with the latter problem. The construction is based on the Riemann existence theorem, exposed, for example, in [6]. Consider a compact connected Riemann surface Σ and a non-constant meromorphic function f on it. The function f can be considered as a ramified covering of the Riemann sphere CP by the surface Σ. Denote by w1, . . . , wk ∈ C the finite ramification points of the covering. (The point [email protected], École Polytechnique, 91128 Palaiseau, France [email protected], bât 425, Université Paris-Sud, 91400 Orsay, France